Factor the following expression: $2$ $x^2$ $-5$ $x$ $-25$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(2)}{(-25)} &=& -50 \\ {a} + {b} &=& & & {-5} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-50$ and add them together. Remember, since $-50$ is negative, one of the factors must be negative. The factors that add up to ${-5}$ will be your ${a}$ and ${b}$ When ${a}$ is ${5}$ and ${b}$ is ${-10}$ $ \begin{eqnarray} {ab} &=& ({5})({-10}) &=& -50 \\ {a} + {b} &=& {5} + {-10} &=& -5 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {2}x^2 +{5}x {-10}x {-25} $ Group the terms so that there is a common factor in each group: $ ({2}x^2 +{5}x) + ({-10}x {-25}) $ Factor out the common factors: $ x(2x + 5) - 5(2x + 5) $ Notice how $(2x + 5)$ has become a common factor. Factor this out to find the answer. $(2x + 5)(x - 5)$